The geometrical origin of the Doppler factor in the Lienard-Wiechert potentials

TL;DR

This paper provides a geometrical derivation of the Doppler factor in Lienard-Wiechert potentials, clarifying its origin and relation to classical Doppler effects through Minkowski space analysis.

Contribution

It introduces a new geometrical interpretation of the Doppler factor for point charges, contrasting it with extended particles, and links it to Fokker's electromagnetic interaction model.

Findings

Doppler factor derived from Minkowski geometry

Distinction between extended and point particles in derivation

Connection between classical and relativistic Doppler effects

Abstract

We present an in depth analysis and a new derivation of the Doppler factor in the Lienard-Wiechert potentials, based on geometrical considerations in Minkowski space. We argue that, contrary to a common assumption, the methods used for deriving the Doppler factor in the case of an electrically charged extended particle are not applicable in the case of a point particle. A geometrical interpretation of the Doppler factor is nonetheless found in the later case, based on the electromagnetic interaction model of Fokker. In this model the interaction takes place between infinitesimal worldline segments with end points connected by light signals, just like the points where light pulses are emitted and detected in general. This analogy reveals that the relativistic Doppler effect is the missing link between the classical Doppler effect and the Doppler factor in the Lienard-Wiechert potentials.